A pentagon with a perimeter of 30 m is to be constructed by

Question

Question 1 -

a. Use the Mean Value Theorem to show that for 0 < x< y,

√y - √x < (y-x)/(2√x).

b. Use part (a), above, to conclude that for 0 < x < y,

√(xy) = (y+x)/2.

Question 2 -

Sketch the graph of a function that satisfies all of the conditions listed below.

a. f(-x) = -f(x)

b. f(0) = 0

c. lim_x→2 f(x) = -∞

d. lim_x→∞f(x) = 0

e. f''(x) < 0 on the intervals (0, 2) and (2, ∞).

Question 3 -

Sketch the graph of the function

f(x) = (2x²-8)/(x²-16).

Question 4 -

Use the closed-interval method to find the absolute maximum and minimum values of the function f(x) = x - 2 sin x on the interval [-π/4, π/2].

Question 5 -

An isosceles triangle has two equal sides of length 10 cm. Let θ be the angle between the two equal sides.

a. Express the area A of the triangle as a function of θ in radians.

b. Suppose that θ is increasing at the rate of 10^o per minute. How fast is A changing at the instant θ = π/3? Item At what value of θ will the triangle haw a maximum area?

Question 6 -

A pentagon with a perimeter of 30 m is to be constructed by adjoining an equilateral triangle to a rectangle. Find the dimensions of the rectangle and triangle that will maximize the area of the pentagon.

Question 7 -

a. Sketch the graphs of the curves y = x and y = cos x showings their point of intersection.

b. Use the Intermediate Value Theorem to identify an interval where the equation cos x = x has a solution.

c. Use Newton's method to approximate the solution of equation cos x = x. Hint: Consider f(x) = x - cos x.